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arXiv:2103.12160 · Journal of Statistical Mechanics Theory and Experiment

Current operators in integrable models: a review

Márton Borsi, Balázs Pozsgay, Levente Pristyák

Published Sep 1, 2021• 55 citations
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Abstract

Abstract We consider the current operators of one-dimensional integrable models. These operators describe the flow of the conserved charges of the models, and they play a central role in generalized hydrodynamics. We present key statements about the mean currents in finite volume and in the thermodynamic limit, and we review the various proofs of the exact formulas. We also present a few new results in this review. New contributions include a computation of the currents of the Heisenberg spin chains using the string hypothesis, and simplified formulas in the thermodynamic limit. We also discuss implications of our results for the asymptotic behavior of dynamical correlation functions.

DOI
10.1088/1742-5468/ac0f6b

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Abstract We consider the current operators of one-dimensional integrable models. These operators describe the flow of the conserved charges of the models, and they play a central role in generalized hydrodynamics. We present key statements about the mean currents in finite volume and in the thermodynamic limit, and we review the various proofs of the exact formulas. We also present a few new results in this review. New contributions include a computation of the currents of the Heisenberg spin chains using the string hypothesis, and simplified formulas in the thermodynamic limit. We also discuss implications of our results for the asymptotic behavior of dynamical correlation functions.
Generalized hydrodynamics is a recent theory that describes the large scale transport properties of one dimensional integrable models. At the heart of this theory lies an exact quantum-classical correspondence, which states that the flows of the conserved quantities are essentially quasiclassical even in the interacting quantum many body models. We provide the algebraic background to this observation, by embedding the current operators of the integrable spin chains into the canonical framework of Yang-Baxter integrability. Our construction can be applied in a large variety of models including the XXZ spin chains, the Hubbard model, and even in models lacking particle conservation such as the XYZ chain. Regarding the XXZ chain we present a simplified proof of the recent exact results for the current mean values, and explain how their quasiclassical nature emerges from the exact computations.
We consider the finite volume mean values of current operators in integrable spin chains with local interactions, and provide an alternative derivation of the exact result found recently by the author and two collaborators. We use a certain type of long range deformation of the local spin chains, which was discovered and explored earlier in the context of the AdS/CFT correspondence. This method is immediately applicable also to higher rank models: as a concrete example we derive the current mean values in the SU(3) SU(3) -symmetric fundamental model, solvable by the nested Bethe Ansatz. The exact results take the same form as in the Heisenberg spin chains: they involve the one-particle eigenvalues of the conserved charges and the inverse of the Gaudin matrix.
We consider the finite volume mean values of current operators in integrable spin chains with local interactions, and provide an alternative derivation of the exact result found recently by the author and two collaborators. We use a certain type of long range deformation of the local spin chains, which was discovered and explored earlier in the context of the AdS/CFT correspondence. This method is immediately applicable also to higher rank models: as a concrete example we derive the current mean values in the SU(3) S U ( 3 ) -symmetric fundamental model, solvable by the nested Bethe Ansatz. The exact results take the same form as in the Heisenberg spin chains: they involve the one-particle eigenvalues of the conserved charges and the inverse of the Gaudin matrix.